It's worth reading the full summary point from the Sapolksy's book, from a chapter titled "Hours to Days Before" [a behavior occurs] — the realm of hormones:

"[...] Within the normal range, individual differences in testosterone levels don't predict who will be aggressive. Moreover, the more an organism has been aggressive, the less testosterone is needed for future aggression. When testosterone does play a role, it's facilitatory—testosterone does not "invent" aggression. It makes us more sensitive to triggers of aggression, particularly in those most prone to aggression. Also, rising testosterone levels foster aggression only during challenges to status. Finally, crucially, the rise in testosterone during a status challenge does not necessarily increase aggression; it increases whatever is needed to maintain status. In a world in which status is awarded for the best of our behaviors, testosterone would be the most prosocial hormone in existence."

Does the exponent of a variable make the variable large or does it describe the characteristics of how it gets large?

I would say the exponent of a variable makes (contributes to making) the result large.

Given that a negative exponent makes a result smaller, I'd say that it describes rate at which it'll change.

What if the base is less than one?

I think the question "does it make it larger or does it describe how it gets large?" implies that the transformation is increasing.

My intention was to say that the exponent describes the characteristics of a change, not to imply that the effects are only positive.